Resources

Resources

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

${:[g(x)=sqrt(2x-9)],[g^(')(x)=?]:} Choose 1 answer: (A) (sqrt(2x-9))/(2) (B) (1)/(sqrt(2x-9)) (C) (1)/(2sqrt(2x-9)) (D) (1)/(sqrtx)$

$\begin{array}{l} g(x)=\sqrt{2 x-9} \\ g^{\prime}(x)=? \end{array}$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{\sqrt{2 x-9}}{2}$ $\newline$ (B) $\frac{1}{\sqrt{2 x-9}}$ $\newline$ (C) $\frac{1}{2 \sqrt{2 x-9}}$ $\newline$ (D) $\frac{1}{\sqrt{x}}$

Full solution

\begin{array}{l} g(x)=\sqrt{2 x-9} \\ g^{\prime}(x)=? \end{array}

\newline

Choose

1

answer:

\newline

(A)

\frac{\sqrt{2 x-9}}{2}

\newline

(B)

\frac{1}{\sqrt{2 x-9}}

\newline

(C)

\frac{1}{2 \sqrt{2 x-9}}

\newline

(D)

\frac{1}{\sqrt{x}}

Identify Functions: First, let's identify the outer function and the inner function. The outer function is the square root function, and the inner function is $(2x-9)$ . We will denote the outer function as $f(u) = \sqrt{u}$ and the inner function as $u(x) = 2x-9$ .
Derivative of Outer Function: Now we need to find the derivative of the outer function $f(u)$ with respect to $u$ . The derivative of $\sqrt{u}$ with respect to $u$ is $\frac{1}{2\sqrt{u}}$ .
Derivative of Inner Function: Next, we find the derivative of the inner function $u(x)$ with respect to $x$ . The derivative of $2x-9$ with respect to $x$ is $2$ .
Apply Chain Rule: Now we apply the chain rule: $g'(x) = f'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $g'(x) = \frac{1}{2\sqrt{2x-9}} \cdot 2$ .
Simplify Expression: Simplify the expression by multiplying the derivatives together. The $2$ from the derivative of the inner function cancels out the $2$ in the denominator of the derivative of the outer function, leaving us with $g'(x) = \frac{1}{\sqrt{2x-9}}$ .